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1.
Ricardo Abreu Blaya Juan Bory Reyes Tania Moreno García 《Bulletin of the Brazilian Mathematical Society》2009,40(1):107-115
In this paper the Théodoresco transform is used to show that, under additional assumptions, each Hölder continuous function f defined on the boundary Γ of a fractal domain Ω ? ?2n can be expressed as f = Ψ+ ? Ψ?, where Ψ± are Hölder continuous functions on Γ and Hermitian monogenically extendable to Ω and to ?2n ? (Ω ∪ Γ) respectively. 相似文献
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3.
Marin Slodi
ka 《Numerical Methods for Partial Differential Equations》2003,19(2):167-191
We study a semilinear parabolic partial differential equation of second order in a bounded domain Ω ? ?N, with nonstandard boundary conditions (BCs) on a part Γnon of the boundary ?Ω. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through Γnon is given, and the solution along Γnon has to follow a prescribed shape function, apart from an additive (unknown) space‐constant α(t). We prove the well‐posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167–191, 2003 相似文献
4.
《Numerical Functional Analysis & Optimization》2013,34(7-8):861-884
We investigate the Cauchy problem for linear elliptic operators with C ∞–coefficients at a regular set Ω ? R 2, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H 1(Ω) solution of an elliptic equation at ?Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically. 相似文献
5.
Ricardo Abreu-Blaya Juan Bory-Reyes 《Bulletin of the Brazilian Mathematical Society》2010,41(3):389-398
Let Ω ? ? n be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum (d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface. 相似文献
6.
Let Ω be a bounded open and oriented connected subset of ? n which has a compact topological boundary Γ, let C be the Dirac operator in ? n , and let ?0,n be the Clifford algebra constructed over the quadratic space ? n . An ?0,n -valued smooth function f : Ω → ?0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F + ? F ? = f on Γ, where the components F ± are extendable to monogenic functions in Ω± with Ω+ := Ω, and Ω? := ? n \ (Ω ? Γ), respectively. 相似文献
7.
Domingo Alberto Tarzia 《Numerical Methods for Partial Differential Equations》1999,15(3):355-369
We consider a material that occupies a convex polygonal bounded domain Ω ⊂ ℝn, with regular boundary Γ = Γ1 ∪ Γ2 (with Γ ∩ Γ = ∅︁) with meas (Γ1) = |Γ1| > 0 and |Γ2| > 0. We assume, without loss of generality, that the melting temperature is 0°C. We consider the following steady‐state heat conduction problem in Ω: with α, q, B = Const > 0, and q and α represent the heat flux on Γ2 and the heat transfer coefficient on Γ1, respectively. In a previous article (Tabacman‐ Tarzia, J Diff Eq 77 (1989), 16– 37) sufficient and/or necessary conditions on data α, q, B, Ω, Γ1, Γ2 to obtain a temperature u of nonconstant sign in Ω (that is, a multidimensional steady‐state, two‐phase, Stefan problem) were studied. In this article, we consider a regular triangulation by finite element method of the domain Ω with Lagrange triangles of the type 1, with h > 0 the parameter of the discretization. We study sufficient (and/or necessary) conditions on data α, q, B, Ω, Γ1, and Γ2 to obtain a change of phase (steady‐state, two‐phase, discretized Stefan problem) in corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Ω. Moreover, error bounds as a function of the parameter h, are also obtained. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq. 15: 355–369, 1999 相似文献
8.
R. Abreu‐Blaya J. Bory‐Reyes M. Shapiro 《Mathematical Methods in the Applied Sciences》2006,29(15):1861-1881
Given a domain Ω in ?3 with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?3 into a sum u = u++u? were u± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
9.
Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains
Hae‐Soo Oh Jae‐Heon Yun Bong Soo Jang 《Numerical Methods for Partial Differential Equations》2006,22(6):1418-1437
The finite element (FE) solutions of a general elliptic equation ?div([aij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R 3, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω∞, so that the remaining part ΩB = Ω\Ω ∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γa = Ω ∞ ∩ Ω B. In this article, instead of discarding an unbounded subdomain Ω∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
10.
Let f be a holomorphic mapping between two bounded domains D and D' in complex space ?n. Suppose that D and D' contain smooth real hypersurfaces Γ and Γ′ as open subsets of their respective boundaries, which correspond under a continuous extension of f. We shall show that this extension is smooth, given certain restrictions on Γ, Γ, and f. 相似文献
11.
Tim Steger 《Milan Journal of Mathematics》1997,67(1):77-85
Let Γ be a free nonabelian group on finitely many generators. Let Ω be the boundary of Γ, letC(Ω) be theC *-algebra of continuous functions on Ω, and let λ be the natural action of Γ onC(Ω). Aboundary representation is a representation of the crossed productC *-algebra Γ×λ C(Ω). Given a unitary representation π of Γ onH, aboundary realization of π is an isometric Γ-inclusion ofH into the space of a boundary representation whose image is cyclic for that boundary representation. If the Γ-inclusion is bijective, we call, the realizationperfect. We prove below that if π admits an imperfect boundary realization, then there exists a nonzero vectorv 0∈H satisfying $$\sum\limits_{|x| = n} {|\left\langle {v,\pi (x)v_0 } \right\rangle |^2 \leqslant |v|^2 } for each v \in {\mathcal{H}} (GVB)$$ If π is irreducible and weakly contained in the regular representation, and if no suchv 0 exists, it follows that π satisfiesmonotony: up to equivalence, there exists exactly one realization of π, and that realization is perfect. 相似文献
12.
Andrew Acker 《Mathematical Methods in the Applied Sciences》1988,10(1):1-14
In References 4 and 5, the author studied the geometric form of the free boundary Γ of an annular domain Ω, characterized by the Bernoulli condition that |?U|=1 on Γ, where U is the capacity potential in Ω. It was shown, for example, that Γ has at most as many ν-extrema (for a given direction ν) or inflection points as the other boundary component Γ* of Ω, which is assumed given. Our purpose here is to extend the results in References 4 and 5 (wherever possible) to multiply connected regions for which some boundary components are given and others are free boundaries characterized by the Bernoulli condition. We present both positive results and counterexamples. 相似文献
13.
E. M. E. Zayed 《Applicable analysis》2013,92(9):905-925
The asymptotic expansions of the trace of the heat kernel for small positive t, where λν are the eigenvalues of the negative Laplacian in Rn (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω1 and a smooth outer boundary ?Ω2 where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ j (j=1,…,m) of ?Ω1 and on the components of ?Ω2 are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given. 相似文献
14.
Ricardo Abreu-Blaya Juan Bory-Reyes Fred Brackx Hennie De Schepper Frank Sommen 《数学学报(英文版)》2012,28(11):2289-2300
A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?2n , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary. 相似文献
15.
We are interested in finding the velocity distribution at the wings of an aeroplane. Within the scope of a three — dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation (Δ +k2)Φ = 0 in ?3\s where Φ denotes the perturbation velocity potential, induced by the presence of the wings and s :=L UW with the projection L of the wings onto the (x,y)- plane and the wake W. Not all Cauchy data are given explicitly on L, respectively W. These missing Cauchy data depend on the wing circulation Γ· Γ has to be fixed by the Kutta–Joukovskii condition: Λ Φ should be finite near the trailing edge xt of L. To fulfil this condition in a way that all appearing terms can be defined mathematically exactly and belong to spaces which are physically meaningful, we propose to fix Γ by the condition of vanishing stress intensity factors of Φ near xt up to a certain order such that ΛΦ|xt ?W2?(xt)? L2(xt),?>0. In the two–dimensional case, and if L is the left half–plane in ?2, we have an explicit formula to calculate Γ and we can control the regularity of Γ and Φ. 相似文献
16.
Gerd Grubb 《Journal of Functional Analysis》1977,26(2):131-165
On a compact n-dimensional C∞ manifold with boundary Γ we consider a matrix A of linear partial differential operators that are not all of the same order. For such systems it is not evident what to regard as Cauchy data (on Γ). We introduce a definition of the so-called reduced Cauchy data, ranging in a vector bundle over Γ, which allows us (1) to set up a Cauchy problem (well posed in the noncharacteristic, analytic case) without the usual extra compatibility condition; (2) to construct boundary projectors in the space of reduced Cauchy data for Douglis-Nirenberg elliptic systems, with which the study of boundary problems is carried over to a (global) study of a pseudodifferential problems over Γ. Further applications (e.g. to spectral theory, outlined in [4]) will be given elsewhere. 相似文献
17.
Uwe Hamann 《Mathematische Nachrichten》1991,154(1):265-284
Let Ω ? Rn be a bounded domain with the smooth boundary Γ, L an elliptic differential operator of order 2m with constant coefficients, E a fundamental solution of L and B = (B1, …, Bm) a normal system of boundary operators on Γ. Furtehr let (yk)Γ1 ? Rn/Ω be a sequence of points and Dj(Γ) (j = 1, …, m) suitable function spaces over Γ (e.g. Cs-spaces or SOBOLEV spaces). It is investigated, under which conditions on the sequence (yk)x1 ? Rn/Ω the set 相似文献
18.
It has been observed13 that the propagation of acoustic waves in the region Ω0= ?2 × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown9 in the case of Dirichlet's boundary condition U = 0 on ?Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω0. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω0 ?B, where B is a smooth bounded domain with B??Ω0. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ? 0 on ? B, where x ′ = (x1, x2, 0) and n is the normal unit vector pointing into the interior B of ? B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω0 will be discussed in a subsequent paper. 相似文献
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20.
Identification of a source factor in a control problem for the heat equation with a boundary memory term 下载免费PDF全文
Cecilia Cavaterra Davide Guidetti 《Mathematical Methods in the Applied Sciences》2015,38(18):4818-4839
We consider a material with thermal memory occupying a bounded region Ω with boundary Γ. The evolution of the temperature u(t,x) is described by an integrodifferential parabolic equation containing a heat source of the form f(t)z0(x). We formulate an initial and boundary value control problem based on a feedback device located on Γ and prescribed by means of a quite general memory operator. Assuming both u and the source factor f are unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution (u,f). Copyright © 2015 John Wiley & Sons, Ltd. 相似文献